The 23-year-old, on loan to the Amsterdam club from Barcelona, sustained the injury early in Ajax's 6-0 thrashing of Go Ahead Eagles at the weekend and could now be out until the middle of November.
An Ajax statement on Tuesday read: "Bojan has a tear in his left hamstring. This was revealed by an MRI scan on Tuesday morning. The attacker is expected to be out for four to six weeks."
Bojan, who has also played for Roma and AC Milan, has returned to Catalonia for treatment.
Ajax lost their Champions League Group H opener 4-0 to Barca a fortnight ago and meet Milan on Tuesday night.
Their form in the Eredivisie has been inconsistent although they lie in fifth place, a point behind joint leaders FC Twente, PSV Eindhoven and Heerenveen.